The expression given is:

\[
\lim_{x \to \frac{\pi}{2}} \sin(x) \cdot x
\]

To evaluate this limit, let's follow these steps:

1. **Substitute** \( x = \frac{\pi}{2} \) directly into the expression.
2. Calculate the sine of \( \frac{\pi}{2} \).
3. Multiply the result by \( \frac{\pi}{2} \).

### Step 1: Substitute \( x = \frac{\pi}{2} \) into the expression

\[
\sin\left(\frac{\pi}{2}\right) \cdot \frac{\pi}{2}
\]

### Step 2: Calculate \( \sin\left(\frac{\pi}{2}\right) \)

\[
\sin\left(\frac{\pi}{2}\right) = 1
\]

### Step 3: Multiply the result by \( \frac{\pi}{2} \)

\[
1 \cdot \frac{\pi}{2} = \frac{\pi}{2}
\]

### Final Answer:

\[
\lim_{x \to \frac{\pi}{2}} \sin(x) \cdot x = \frac{\pi}{2}
\]

Would you like further details or have any more questions?

Here are five related questions for further exploration:

1. What is the limit of \( \sin(x) \) as \( x \) approaches other special angles?
2. How does the behavior of trigonometric functions influence limits?
3. What happens if the angle approaches an undefined point for \( \sin(x) \)?
4. Can limits involving trigonometric functions be evaluated graphically?
5. How does the unit circle help in understanding sine and cosine functions?

**Tip:** For trigonometric limits, understanding the basic properties of sine and cosine at key angles (like \( 0, \frac{\pi}{2}, \pi \)) is very helpful.

Learn how to evaluate the limit of sin(x) * x as x approaches π/2 with a detailed step-by-step solution. This problem involves limits and trigonometry, focusing on the properties of the sine function and its behavior near π/2. Suitable for advanced high school students and college-level calculus enthusiasts.

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